Discussion
Let X and Y be random variable with the following distributions respectively :
| X | 1 | 2 | 3 | 4 | 5 |
| P | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
| Y | 10 | 20 | 35 | 50 | 60 |
| P | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
Answer: C
The entropies are the same, but the coefficients of variation are not.
Entropy depends only on the probabilities, \(H = -\sum p_i \log p_i\), not on the values the variable takes. X and Y have the same probabilities (0.1, 0.2, 0.4, 0.2, 0.1), so their entropies are equal.
The coefficient of variation is the standard deviation divided by the mean, and it depends on the values:
\[ \begin{aligned} \mu_X &= 3,\ \sigma_X^2 = 1.2,\ CV_X \approx 0.365 \\ \mu_Y &= 35,\ \sigma_Y^2 = 215,\ CV_Y \approx 0.419 \end{aligned} \]
Y's values (10, 20, 35, 50, 60) are not a fixed multiple of X's, since 10X would give 10 to 50, so the CVs differ. Options saying the entropies differ are wrong, and so is the option saying the CVs match.
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